PS1Sol - Econ104(3)

PS1Sol - Econ104(3) - Econ 104 Problem Set 1 Solutions...

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Unformatted text preview: Econ 104 - Problem Set 1 Solutions Lorenzo Braccini * September 22, 2011 Problem 1 First note that if X and Y represent respectively the salary of a scientist in thousands of dollars and in dollars, when X = x we have that Y = 1000 x . This is true for any x in the range of X , hence: Y = 1000 X a) By standard properties of the expectation operator we have that: E [ Y ] = E [1000 X ] = 1000 E [ X ] = 1000 × 48 . 8 = 48 , 800$ b) Again by the definition of standard deviation and properties of the variance of a random variable we can write: σ Y = q σ 2 Y = p V ar ( Y ) = p V ar (1000 X ) = p 1000 2 V ar ( X ) = 1000 p V ar ( X ) = 1000 σ X = 12 , 100$ Problem 2 Let X be a Bernoulli distributed Random Variable with parameter p . Define Z = 3 X- 1. a) Yes, Z is a Random Variable. In fact, it takes values on the set * [email protected] 1 { , 2 } with probability: Pr ( Z = 0) = Pr (3 X- 1 = 0) = Pr (3 X = 1) = Pr ( X = 0) = 1- p and Pr ( Z = 2) = Pr (3 X- 1 = 2) = Pr (3 X = 3) = Pr ( X = 1) = p b) Again by the standard properties of the expectation operator we obtain that: E [ Z ] = E 3 X- 1 = E 3 X- 1 Also by the definition of the expectation we have: E 3 X = X x ∈{ , 2 } 3 x Pr ( X = x ) = 1(1- p ) + 3 p = 2 p + 1 Hence: E [ Z ] = 2 p + 1- 1 = 2 p c) By a similar reasoning we have: E Z 2 = E (3 X- 1) 2 = E 3 2 X- 2 × 3 X + 1 = E...
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PS1Sol - Econ104(3) - Econ 104 Problem Set 1 Solutions...

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